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4Pythagorean triples a b c are positive integer solutions to the Pythagorean theorem a² b² c² Some simple rules about Pythagorean theorem are as following Every odd number is the a side for the Pythagorean triples The formula of the b side is a² 1 2 The formula of the c side is b 1 This report will discuss how Pythagorean triples work in details and how teachers can use this to create more questions using whole numbers and perfect squares Rationale Pythagorean theorem is one of the most famous theorems It has a lot of applications in trigonometry however it is sometimes annoying to use it when finding the side length of a triangle and getting an irrational number It is rather satisfactory when finding a positive integer My partner and I had often wondered if there was a way for teachers to always give us questions in which the answer would be a positive integer Upon some research we found that there is in fact a way to make sure the solutions to these kind of questions will result in a positive integer It is through the use of pythagorean triples In this internal assessment IA we will be discussing what the pythagorean triple is its formula and how teachers can use this to create more questions using whole numbers and perfect squares Some key concepts you will need to know to understand this IA is coprime meaning mutually prime primitive infinite number of solutions Description Dickson s method

This IA will be discussing how the formula can be used to find pythagorean triple In 1920 Leonard Eugene Dickson found that to find integer solutions to find positive integers r s and t such that Then From this it is clear that is any even integer We can also see that s and t are factors of When s and t are coprime the triple value of r s and t will be primitive Here is the proof for this formula Example If we chose r 6 and plug it into Then we find that This results in the three factor pairs of 18 which are 1 18 2 9 and 3 6 These three factor pairs will now create triples using the equation stated above First Factor Pair s 1 t 18 produces 7 24 25 This happens because x 6 1 7 y 6 18 24 z 6 1 18 25 Second Factor Pair s 2 t 9 produces 8 15 17 This happens because x 6 2 8 y 6 9 15 z 6 2 9 17 Third Factor Pair s 3 t 6 produces 9 12 15 This happens because x 6 3 9 y 6 6 12 z 6 3 6 15 Since s and t are not coprime this triple is not primitive Below is a table showing the data that we got using the formulas mentioned before for the Pythagorean triples a b and c a is the independent variable that could be any odd number on the positive side of the number line b and c are dependent variables that varies their values based on the value of a a 1 3 5 7 9 11 13 15 17 19 21 23 25 27 b 4 12 24 40 60 84 122 144 180 220 264 312 364 c 1 5 13 25 41 61 85 123 145 181 221 265 313 365 Notice that a and c are always odd and b is always even These patterns work because the differences between consecutive square numbers are consecutive odd numbers For example the square of 7 which is the difference between the square of 24 which is 576 and the square of 25 which is 625 giving us the triplet 7 24 25

The fact that the differences between consecutive square numbers are consecutive odd numbers lead us to this Pythagorean discovery Every square is the sum of two consecutive triangular numbers where the triangular numbers are the consecutive sums of all integers 0 1 1 0 1 2 3 0 1 2 3 6 etc So the triangular numbers are 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 etc 105 120 225 225 is the square of 15 The way this relationship can be illustrated is that as every square can be divided into two triangles so can every square number It can be divided into two triangular numbers which can be mapped as triangles The square 25 can then be divided into the triangles 10 and 15 This kind of relationship will thus help teachers in different fields to create more questions using whole numbers and perfect squares Conclusion In conclusion we can say that teachers can in fact use this formula to find pythagorean triples to use for their student s work We found through this IA that pythagorean triple is integer solutions to the pythagorean theorem Pythagorean triples formula is which we can find with the use of this finds the factor Lastly we found that teachers can use this formula by substituting any value into the formula as explained above and create more questions using whole numbers and perfect squares

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